A spring is mounted vertically, with the lower end at 100 cm from the floor. Aft
ID: 581812 • Letter: A
Question
A spring is mounted vertically, with the lower end at 100 cm from the floor. After a 500 g block is connected to the spring, it gets the equilibrium position at 97.0 cm from the floor. The block undergoes simple harmonic motion after it is displaced 30.0 mm form the position of equilibrium, and then released. Calculate the force constant of the spring. Calculate and define period, frequency and angular frequency of the resulting oscillation. Calculate the maximum speed and displacement (related to the floor) experienced by the block. Calculate the total energy (mechanical energy) of the block at the equilibrium position and when the block has it maximum displacement. Write the equation that describe the motion of the spring. Why we can state that the motion of the block corresponds to a simple harmonic motion?Explanation / Answer
a) F = kx
=> mg = kx
=> k = 0.5 * 9.8/0.03
= 163.33 N/m
b) Angular frequency is the number of radians of the oscillation that are completed each second.
= sqrt(k/m)
= sqrt(163.33/0.5)
= 18.07 rad/sec
frequency, f, tells the number of full cycles per second .
f = 18.07/(2*3.14)
= 2.877 Hz
Time period is time taken to complete a full cycle .
T = 1/2.877
= 0.347 sec
c) maximum speed = 18.07 * 0.03
= 0.5421 m/sec
maximum displacement = 100 cm
d) total energy at equilibrium position = 0.5 * 163.33 * 0.03 *0.03
= 0.0735 J
total energy at maximum displacement = 0.5 * 163.33 * 0.03 *0.03
= 0.0735 J
e) Equation that describe spring motion
=> y = 0.03 cos(18.07t)
f) All simple harmonic motion is sinusoidal. A mass on a spring undergoing simple harmonic motion slows down at the very top and bottom, before gradually increasing speed again as it approaches the center. It spends more time at the top and bottom than it does in the middle. Any motion that has a restoring force proportional to the displacement from the equilibrium position will vary in this way.
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